Generalized cylindrical cavity system for microwave rotation and impedance shifting by irises in a power-supplying waveguide

ABSTRACT

A rotating microwave is established for any resonant mode TE mnl  or TM mnl  of a cavity, where the user is free to choose the values of the mode indices m, n and 1. The fast rotation, the rotation frequency of which is equal to an operational microwave frequency, is accomplished by setting the temporal phase difference ΔØ and the azimuthal angle Δθ between two microwave input ports P and Q as functions of m, n and 1. The slow rotation of frequency Ω α  (typically 1-1000 Hz), is established by transforming dual field inputs α cos Ω α t and ±α sin Ω α t in the orthogonal input system into an oblique system defined by the angle Δθ between two microwave ports P and Q.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/247,955, filed Oct. 29, 2015 entitled GENERALIZED CYLINDRICAL CAVITYSYSTEM FOR MICROWAVE ROTATION AND IMPEDANCE SHIFTING BY IRISES IN APOWER-SUPPLYING WAVEGUIDE, by Satoru Kobayashi, et al.

BACKGROUND

Technical Field

The disclosure concerns the production of rotating microwaves in aplasma reactor chamber.

Background Discussion

In one approach for generating rotating microwaves in a plasma reactorchamber, microwaves are radiated into a cylindrical cavity from twoports separated spatially 90 degree. By setting a temporal phasedifference between the microwaves from the two ports at 90 degrees, theTE₁₁₁ mode in a cylindrical cavity is rotated circularly with feedbackcontrol by two monitoring antennas, providing a plasma profile of highuniformity.

In another approach for generating rotating microwaves, temporal phasesbetween the two microwaves radiated from the two ports are kept inphase. To produce rotation, an amplitude of the microwaves from one portis modulated in the form of A sin Ωt, while an amplitude of microwavesfrom the other port is modulated in the form of A cos Ωt. Here, Ω is anangular frequency of order of 1-1000 Hz, which is much smaller than thatof a microwave carrier frequency of order of over 1 GHz. This dualinjection rotates the TE₁₁₁ mode at a slow frequency Ω so as to slowlyagitate a localized plasma, spreading the plasma into a wider area tofurther increase a uniforminy of plasma distribution, particularly athigh pressures.

However, the fast and slow rotations were provided only for the TE₁₁₁mode. There is a need to provide such rotation for any mode, not justthe TE₁₁₁ mode.

SUMMARY

In a plasma reactor comprising a cylindrical microwave cavity overlyinga workpiece processing chamber, and first and second microwave inputports P and Q in a sidewall of the cylindrical microwave cavity spacedapart by an offset angle Δθ, a method is provided for generatingrotating microwaves of mode TE_(mnl) or TM_(mnl) in the cylindricalmicrowave cavity, wherein n, m and 1 are user-selected values of a TE orTM mode. The method comprises: introducing into the cylindricalmicrowave cavity, through respective ones of the first and secondcoupling apertures, respective microwave signals separated by a temporalphase difference ΔØ; adjusting values of the offset angle Δθ and thetemporal phase difference ΔØ to values which are a function of at leasttwo of the user-selected TE or TM mode indices m, n and 1 so as toproduce rotating microwaves of mode TE_(mnl) or TM_(mnl) in thecylindrical microwave cavity.

In one embodiment, the function is defined as:

$\quad\left\{ \begin{matrix}{\frac{{m\; {\Delta\theta}} - {\Delta \; \varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k\text{:}\mspace{14mu} {integer}} \\{{\Delta\varphi} \neq {p\; \pi}} & {p\text{:}\mspace{14mu} {integer}}\end{matrix} \right.$

In one embodiment, the rotating microwaves rotate clockwise with therotation frequency equal to an operational microwave frequency.

In one embodiment, to maximize the energy transfer efficiency of theclockwise rotation, the function is defined as:

$\quad\left\{ \begin{matrix}{\frac{{m\; {\Delta\theta}} - {\Delta \; \varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k\text{:}\mspace{14mu} {integer}} \\{\frac{{m\; {\Delta\theta}} + {\Delta \; \varphi}}{2} = {p\; \pi}} & {p\text{:}\mspace{14mu} {integer}}\end{matrix} \right.$

In one embodiment, the function is defined as:

$\quad\left\{ \begin{matrix}{\frac{{m\; \Delta \; \theta} + {\Delta \; \varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k\text{:}\mspace{11mu} {integer}} \\{{\Delta \; \varphi} \neq {p\; \pi}} & {p\text{:}\mspace{11mu} {integer}}\end{matrix} \right.$

In one embodiment, the rotating microwaves rotate anticlockwise with therotation frequency equal to an operational microwave frequency.

In one embodiment, to maximize the energy transfer efficiency of theanticlockwise rotation, the function is defined as:

$\quad\left\{ \begin{matrix}{\frac{{m\; \Delta \; \theta} + {\Delta \; \varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k\text{:}\mspace{11mu} {integer}} \\{\frac{{m\; \Delta \; \theta} - {\Delta \; \varphi}}{2} = {p\; \pi}} & {p\text{:}\mspace{11mu} {integer}}\end{matrix} \right.$

In one embodiment, a first one of the respective microwave signals is ofa form:

H_(P) ∝ cos(η+mθ−ωt)+cos(η+mθ+ωt)

where ω is an angular frequency of the respective microwave signals andt is time, and η=0 or

$\frac{\pi}{2}.$

In one embodiment, a second one of the respective microwave signals isof a form:

H_(Q)∝ cos[η+m(θ−Δθ)−(ωt−Δφ)]+ cos[η+m(θ−Δθ)+(ωt−Δφ)]

where ω is an angular frequency of the microwave signals and t is time,and η=0 or

$\frac{\pi}{2}.$

In a plasma reactor comprising a cylindrical microwave cavity overlyinga workpiece processing chamber, and first and input ports in a sidewallof said cylindrical microwave cavity spaced apart by a general angle, amethod is provided for generating rotating microwaves in saidcylindrical microwave cavity with rotation frequency Ω_(α), the methodcomprising:

setting said general angle to satisfy the following equations:

â=a _(x) {circumflex over (x)}+a _(y) ŷ

{circumflex over (b)}=b _(x) {circumflex over (x)}+b _(y) ŷ;

inputting to input ports P and Q microwave fields representedrespectively by:

ζ_(Pa) =r cos(ωt+φ_(h))

ζ_(Qb) =s cos(ωt+φ_(h))

where r and s are defined in the following equations:

$r = \frac{{\alpha \mspace{14mu} \cos \; \Omega_{a}t\mspace{14mu} b_{y}} \mp {\alpha \mspace{14mu} \sin \; \Omega_{a}t\mspace{14mu} b_{x}}}{V}$$s = \frac{{{- \alpha}\mspace{14mu} \cos \; \Omega_{a}t\mspace{14mu} a_{y}} \pm {\alpha \mspace{14mu} \sin \; \Omega_{a}t\mspace{14mu} a_{x}}}{V}$

and the sign ∓ in “r” determines whether the rotation is anticlockwiseor clockwise.

In accordance with a further aspect, a plasma reactor comprises: acylindrical microwave cavity overlying a workpiece processing chamber,and first and second input ports, P and Q, in a sidewall of thecylindrical microwave cavity spaced apart by an azimuthal angle; amicrowave source having a microwave frequency and having a pair ofmicrowave source outputs; a pair of respective waveguides, each of therespective waveguides having a microwave input end coupled to arespective one of the microwave source outputs and a microwave outputend coupled to a respective one of the first and second input ports; acoupling aperture plate at the output end, and a rectangular couplingaperture in the coupling aperture plate; an iris plate between thecoupling aperture plate and the microwave input end, and a rectangulariris opening in the iris plate.

In one embodiment, the rectangular coupling aperture and the rectangulariris opening have respective parallel axes along a long dimension of arespective one of the coupling aperture and the iris opening, therespective parallel axes being parallel to an axis of symmetry of thecylindrical microwave cavity.

In one embodiment, each of the waveguides has a microwave propagationdirection between the microwave input end and the microwave output end,the microwave propagation direction extending toward an axis of symmetryof the cylindrical microwave cavity.

In one embodiment, the rectangular coupling aperture has long and shortdimensions e and f, respectively, corresponding to a user-selectedimpedance.

In one embodiment, the rectangular iris opening has long and shortdimensions c and d, respectively, corresponding to a user-selectedresonance.

In one embodiment, the rectangular iris is a capacitive iris and has along dimension parallel to an axis of symmetry of the cylindricalmicrowave cavity.

In one embodiment, the rectangular iris is an inductive iris and has ashort dimension parallel to the axis of symmetry of the cylindricalmicrowave cavity.

BRIEF DESCRIPTION OF THE DRAWINGS

So that the manner in which the exemplary embodiments of the presentinvention are attained can be understood in detail, a more particulardescription of the invention, briefly summarized above, may be had byreference to the embodiments thereof which are illustrated in theappended drawings. It is to be appreciated that certain well knownprocesses are not discussed herein in order to not obscure theinvention.

FIG. 1A is cross-sectional elevational view of a plasma reactor that maybe used in carrying out embodiments.

FIG. 1B is a plan view corresponding to FIG. 1A.

FIG. 1C is a plan view of a related reactor.

FIG. 2 is a schematic diagram of a system including the reactor of FIG.1A.

FIG. 3 is a schematic diagram of another system including the reactor ofFIG. 1A.

FIG. 3A is a block diagram depicting a method of operating the system ofFIG. 3.

FIGS. 4 and 5 depict coordinate systems referred to in the detaileddescription.

FIG. 6 is a diagram of a system including a pair of power-feedingwaveguides having impedance-shifting irises.

FIG. 6A is a plan view corresponding to FIG. 6.

FIGS. 7, 8 and 9 depict different irises for use in each waveguide ofFIG. 6.

To facilitate understanding, identical reference numerals have beenused, where possible, to designate identical elements that are common tothe figures. It is contemplated that elements and features of oneembodiment may be beneficially incorporated in other embodiments withoutfurther recitation. It is to be noted, however, that the appendeddrawings illustrate only exemplary embodiments of this invention and aretherefore not to be considered limiting of its scope, for the inventionmay admit to other equally effective embodiments.

DETAILED DESCRIPTION Introduction:

In the present description, microwave field rotations are provided forthe general case of TE_(mnl) and TM_(mnl) in a cylindrical cavity, wherem, n and 1 are suitable integers chosen by the user. Our recentexperimental work confirms that a TE₁₂₁ mode makes a higher uniformityof plasma distribution under some conditions.

In addition, a method for changing a chamber impedance by using irisesinstalled in a power-supplying waveguide is disclosed. In general, acylindrical cavity has a bottom plate on which radiation slots are cutout to transfer microwave energy from the cavity to plasma. For a givendesign of the radiation slots, the chamber impedance is fixed. If thechamber impedance is in a region controlled by a stub tuner, thestub-tuner will make an impedance matching easily. Otherwise, the tuningbecomes unpredictable or unstable to make an oscillation of the tuningposition. Inversely, if the chamber impedance is controlled, it can bemoved to tuner-preferred regions, which further leads to reduction ofnumbers of stubs, leading to cost reduction. The method proposed hereinis simple, and moves the chamber impedance into wide ranges in the Smithchart as demonstrated in our recent experiments. The adoption of thismethod will provide stable plasma tuning and chamber-to-chamberetching/plasma matching.

Fast Rotation of TE_(mnl) and TM_(mnl) Modes in a Cylindrical Cavitywith a Microwave Carrier Frequency:

In this description, fast rotation is defined as a field rotation withthe same rotation frequency as an operational microwave frequency. FIG.1A is a simplified side view of a plasma reactor 100 including aprocessing chamber 110 enclosed by a wall 111 and containing gas undervacuum pressure and a workpiece support 112 for supporting a workpiece114. A cylindrical cavity 120 overlying the processing chamber 110 isenclosed by a side wall 121 a, a ceiling 121 b and a floor 122 havingslots 124 shown in FIG. 1B. The walls 121 a and 111 can be connected bymetal structures, depending upon application. A dielectric plate 130provides a vacuum seal under the floor 122. The dielectric plate 130 ispreferably formed of a material that is transparent to microwaveradiation. FIG. 1C depicts an embodiment in which the floor 122 has anopening 810 and an auxiliary ignition electrode 820 is disposed in theopening 810 with a vacuum seal (not shown). The auxiliary ignitionelectrode 820 is driven by an RF source 830 of an RF frequency in arange of 100 Hz-10 MHz. The RF source 830 may include an impedance match(not illustrated). The floor 122 and/or the wall 111 of the processingchamber 110 can function as a ground plane with respect to the auxiliaryignition electrode 820. Alternatively, an auxiliary ignition electrodecan be located on the wall 111 by providing an additional opening andvacuum seal. The electrode 820 and the ground plane are separated onlyby the opening 810. In summary, the auxiliary ignition electrode 820together with the ground plane (i.e., the floor 122 and/or the wall 111of the chamber 110) form a capacitively coupled RF igniting circuit tohelp ignition of plasma that is ultimately sustained by microwave power.

FIG. 2 depicts an embodiment in which first and second microwave inputports, P and Q, in the side wall 121 a are located at azimuthalpositions spaced apart by a non-orthogonal angle relative to oneanother. In FIG. 2, two identical microwave modules, Set-1 and Set-2,are connected to the cylindrical cavity 120 at input port P (where θ=0),and input port Q (where θ=θ_(q)), respectively. The other ends of themodules, Set-1 and Set-2, are connected to respective output signals,A₁and A₂, of a dual digital seed (phase and amplitude) generator 340that supplies microwave signals to the modules Set 1 and 2. In eachmodule, the seed signal is amplified by a solid-state amplifier 350,which transmits it to a circulator 352 and a tuner 354, typically 3-polestub tuner, to reduce reflection. The microwave is finally introducedinto the cylindrical cavity 120 through a waveguide 360 with a radiatingaperture, and excites eigen modes (resonances) in the cylindrical cavity120. In general, transmission lines are used from the output of theamplifier 350 to the stub tuner 354. In this example, the radiatingaperture is placed at the tip of the waveguide 360. Acoaxial-to-waveguide transformer 358 is inserted between the tuner 354and the waveguide 360. However, if a pole or loop antenna is adopted,the transformer 358 can be removed. In addition, a dummy load 362 isconnected into one end of the circulator 352 to protect the amplifier350.

Monitoring antennas 200 a and 200 b are orthogonally placed to receivemicrowave signals. The signal received by each one of the monitoringantennas 200 a and 200 b is processed by a signal feedback controller340-1. In the feedback controller 340-1, the in- and quadrature-phasedemodulation (IQ demodulation) is performed to measure the phase andamplitude of the received signal at the microwave frequency. When thisphase and amplitude detection is performed for both the modules, Sets 1and 2, the controller 340-1 calculates the mutual temporal phasedifference Δφ and the amplitudes of the output signals, A₁ and A₂ usingdigital signal processing. Since the circularly fast rotation ofTE_(mnl) and TM_(mnl) mode in a cylindrical cavity with a microwavecarrier frequency requires Δφ=±90° and A₁=A₂, the controller 340-1performs feedback-loop control, until the required relation issatisfied. This feed-back is operated independently from stub tuningworks. Hence, as long as high speed controllers, such as an FPGA and amicrocontroller, are used, a prompt conversion to the required conditionis achieved in less than a millisecond.

Representation of Electromagnetic Fields of TE_(mnl) in a ResonantCavity:

In FIG. 2, the angled (non-orthogonal) orientation of the input ports Pand Q requires a new condition on a temporal phase delay Δφ of Q fromthat of P to make the circular fast rotation. As already stated, thefeedback monitoring system can take care of control to make the perfectcircular fast rotation. However, it is desirable to set a best initialvalue of Δφ so that feedback control time becomes minimized. Δφ is nowderived for TE_(mnl) and TM_(mnl). In the following, h=height of acylindrical cavity, and R=radius of the cylindrical cavity.

For TE_(mnl), the fields are represented for given a single integer ‘m’in Gauss units:

$\begin{matrix}{{{\kappa^{2} = {{\frac{\omega^{2}}{c^{2}}ɛ} - k_{zl}^{2}}}B_{z} = {{J_{m}\left( {\kappa \; r} \right)}{^{\; m\; \theta}\left( {{A_{+}^{\; k_{zl}z}} + {{A\_}\; ^{{- }\; k_{zl}z}}} \right)}}}{\overset{\rightarrow}{B_{t}} = {\frac{\; k_{zl}}{\kappa^{2}}{^{\; m\; \theta}\left\lbrack {{\kappa \; {J_{m}^{\prime}\left( {\kappa \; r} \right)}\hat{r}} + {\frac{\; m}{r}{J_{m}\left( {\kappa \; r} \right)}\hat{\theta}}} \right\rbrack}\left( {{A_{+}^{\; k_{zl}z}} - {A_{\_}^{{- }\; k_{zl}z}}} \right)}}{\overset{\rightarrow}{E_{t}} = {\frac{\; \omega}{c\; \kappa^{2}}{^{\; m\; \theta}\left\lbrack {{\frac{\; m}{r}{J_{m}\left( {\kappa \; r} \right)}\hat{r}} - {\kappa \; {J_{m}^{\prime}\left( {\kappa \; r} \right)}\hat{\theta}}} \right\rbrack}{\left( {{A_{+}^{\; k_{zl}z}} + {A\_ }^{{- }\; k_{zl}z}} \right).}}}} & (1)\end{matrix}$

The boundary condition that the tangential components of electric fieldsin the cavity must vanish leads to the following relations:

$\begin{matrix}{{A_{+} = {{{- A}\; \_} \equiv \frac{\; A}{2}}}{k_{zl} = {\frac{l\; \pi}{h}\mspace{14mu} l\text{:}\mspace{11mu} {integer}}}{\kappa = {y_{mn}^{\prime}\text{/}R}}{{{where}\mspace{14mu} {J_{m}^{\prime}\left( y_{mn}^{\prime} \right)}} = 0.}} & (2)\end{matrix}$

Then, the fields become

$\begin{matrix}{{B_{z} = {{{AJ}_{m}\left( {\kappa \; r} \right)}^{\; m\; \theta}{\sin \left( {k_{zl}z} \right)}}}{\overset{\rightarrow}{B_{t}} = {A\frac{k_{zl}}{\kappa^{2}}{^{\; m\; \theta}\left\lbrack {{\kappa \; {J_{m}^{\prime}\left( {\kappa \; r} \right)}\hat{r}} + {\frac{\; m}{r}{J_{m}\left( {\kappa \; r} \right)}\hat{\theta}}} \right\rbrack}\cos \; \left( {k_{zl}z} \right)}}{\overset{\rightarrow}{E_{t}} = {A\frac{\omega}{c\; \kappa^{2}}{^{\; m\; \theta}\left\lbrack {{\frac{- m}{r}{J_{m}\left( {\kappa \; r} \right)}\hat{r}} - {{\kappa}\; {J_{m}^{\prime}\left( {\kappa \; r} \right)}\hat{\theta}}} \right\rbrack}{{\sin \left( {k_{zl}z} \right)}.}}}} & (3)\end{matrix}$

When considering two degenerate ‘n’ and ‘−n’ along with the temporalterm e^(−iωt), we can write the magnetic fields as:

$\begin{matrix}{\left. {{B_{z} = {A\; {J_{m}\left( {\kappa \; r} \right)}^{\; m\; \theta}{{\sin \left( {k_{zl}z} \right)}\left\lbrack {{a\mspace{14mu} {\cos \left( {{m\; \theta} - {\omega \; t}} \right)}} + {b\mspace{14mu} {\cos \left( {{m\; \theta} + {\omega \; t}} \right)}}} \right\rbrack}}}{B_{r} = {A\; \frac{k_{zl}}{\kappa}{J_{m}^{\prime}\left( {\kappa \; r} \right)}\; {{\cos \left( {k_{zl}z} \right)}\left\lbrack {{a\mspace{14mu} {\cos \left( {{m\; \theta} - {\omega \; t}} \right)}} + {b\mspace{14mu} {\cos \left( {{m\; \theta} + {\omega \; t}} \right)}}} \right\rbrack}}}} \right){B_{\theta} = {{- A}\; \frac{m\; k_{zl}}{\kappa^{2}r}{J_{m}^{\prime}\left( {\kappa \; r} \right)}\; {{\cos \left( {k_{zl}z} \right)}\left\lbrack {{a\mspace{14mu} {\sin \left( {{m\; \theta} - {\omega \; t}} \right)}} + {b\mspace{14mu} {\sin \left( {{m\; \theta} + {\omega \; t}} \right)}}} \right\rbrack}}}} & (4)\end{matrix}$

where a and b are constants.

All the magnetic field components at a fixed (r, z) can be written withnewly normalized constants a and b in the form of

B=a cos(η+mθ−ωt)+b cos(η+mθ+ωt)   (5)

where η=0 or

$\frac{\pi}{2}.$

Specifically, in Eqn. (5), “a” and “b” are amplitude coefficients of theanticlockWISE and clockwise rotation, respectively.

Representation of Electromagnetic Fields of TM_(mnl) in a ResonantCavity:

For TM_(mnl), the fields are represented for given a single integer, ‘m’in Gauss units:

$\begin{matrix}{{\kappa^{2} = {{\frac{\omega^{2}}{c^{2}}ɛ} - k_{zl}^{2}}}{E_{z} = {{J_{m}\left( {\kappa \; r} \right)}{^{\; m\; \theta}\left( {{A_{+}^{\; k_{zl}z}} + {A\_ }^{{- }\; k_{zl}z}} \right)}}}{\overset{\rightarrow}{E_{t}} = {\frac{\; k_{zl}}{\kappa^{2}}{^{\; m\; \theta}\left\lbrack {{\kappa \; {J_{m}^{\prime}\left( {\kappa \; r} \right)}\hat{r}} + {\frac{\; m}{r}{J_{m}\left( {\kappa \; r} \right)}\hat{\theta}}} \right\rbrack}\left( {{A_{+}^{\; k_{zl}z}} - {A\_ }^{{- }\; k_{zl}z}} \right)}}{\overset{\rightarrow}{B_{t}} = {{- \frac{\; \omega \; ɛ}{c\; \kappa^{2}}}{^{\; m\; \theta}\left\lbrack {{\frac{\; m}{r}{J_{m}\left( {\kappa \; r} \right)}\hat{r}} - {\kappa \; {J_{m}^{\prime}\left( {\kappa \; r} \right)}\hat{\theta}}} \right\rbrack}{\left( {{A_{+}^{\; k_{zl}z}} + {A\_ }^{{- }\; k_{zl}z}} \right).}}}} & (6)\end{matrix}$

In a similar manner to TE_(mnl), the boundary condition that thetangential components of electric fields in the cavity must vanish leadsto the following relations with slight changes

$\begin{matrix}{{{A_{+} = {{A\_} \equiv \frac{A}{2}}}{k_{zl} = {\frac{l\; \pi}{h}\mspace{14mu} l\text{:}\mspace{11mu} {integer}}}\text{}{\kappa = {y_{mn}\text{/}R}}}{{{where}\mspace{14mu} {J_{m}\left( y_{mn} \right)}} = 0.}} & (7) \\{{E_{z} = {A\; {J_{m}\left( {\kappa \; r} \right)}^{\; m\; \theta}{\cos \left( {k_{zl}z} \right)}}}{\overset{\rightarrow}{E_{t}} = {{- A}\; \frac{k_{zl}}{\kappa^{2}}{^{\; m\; \theta}\left\lbrack {{\kappa \; {J_{m}^{\prime}\left( {\kappa \; r} \right)}\hat{r}} + {\frac{\; m}{r}{J_{m}\left( {\kappa \; r} \right)}\hat{\theta}}} \right\rbrack}{\sin \left( {k_{zl}z} \right)}}}{\overset{\rightarrow}{B_{t}} = {A\; \frac{\omega \; ɛ}{c\; \kappa^{2}}{^{\; m\; \theta}\left\lbrack {{\frac{m}{r}{J_{m}\left( {\kappa \; r} \right)}\hat{r}} + {\; \kappa \; {J_{m}^{\prime}\left( {\kappa \; r} \right)}\hat{\theta}}} \right\rbrack}\cos \; {\left( {k_{zl}z} \right).}}}} & (8)\end{matrix}$

When considering both n and -n along with the temporal term e^(−iωt), wecan write the magnetic fields as

$\begin{matrix}{\mspace{79mu} {{B_{z} = 0}{B_{r} = {{- A}\; \frac{\omega \; ɛ\; m}{c\; \kappa^{2}r}{J_{m}\left( {\kappa \; r} \right)}{{\cos \left( {k_{zl}z} \right)}\left\lbrack {{a\mspace{14mu} {\cos \left( {{m\; \theta} - {\omega \; t}} \right)}} + {b\mspace{14mu} {\cos \left( {{m\; \theta} + {\omega \; t}} \right)}}} \right\rbrack}}}\mspace{79mu} {B_{\theta} = {A\; \frac{\omega \; ɛ}{c\; \kappa \mspace{14mu} r}{J_{m}^{\prime}\left( {\kappa \; r} \right)}{{{\cos \left( {k_{zl}z} \right)}\left\lbrack {{a\mspace{14mu} {\sin \left( {{m\; \theta} - {\omega \; t}} \right)}} + {b\mspace{14mu} {\sin \left( {{m\; \theta} + {\omega \; t}} \right)}}} \right\rbrack}.}}}}} & (9)\end{matrix}$

All the magnetic field components at a fixed (r, z) can be written withnewly normalized constants a and b in the form of:

B=a cos(η+mθ−ωt)+b cos(η+mθ+ωt)   (10)

where η=0 or

$\frac{\pi}{2}.$

Since Eqn. (10) is of identical form to Eqn. (5), the followingdiscussions can be applied to both TE_(mnl) and TM_(mnl). For the sakeof brevity, the term η in Eqns. (5) and (10) will be dropped in thefollowing discussion.Single and Dual Injection for TE_(mnl) and TM_(mnl):

When considering wave excitation from Port P, anticlockwise andclockwise rotations are excited with equal probabilities as a firstapproximation. Then, the excited wave can be written by renormalizingthe coefficients a and b in Eqn. (2) as unity:

H _(p)=cos(mθ−ωt)+cos(mθ+ωt).   (11)

Next, when exciting a wave from Port Q with the same power andfrequency, however, with a temporal phase delay of Δφ, the excited wavecan be represented as:

H _(Q)=cos[m(θ−Δθ)−(ωt−Δφ)]+cos[m(θ−Δθ)+(ωt−Δφ)]  (12)

where Δθ is the angular offset in position of Port Q relative to Port P,and Δφ is the temporal phase difference between the microwave outputs A₁and A₂. When exciting the cavity 120 from both input ports P and Qsimultaneously, the excited wave can be given as a sum of Eqns. (11) and(12):

H_(tot)=cos(mθ−ωt)+cos(mθ+ωt)+cos[m(θ−Δθ)−(ωt−Δφ)]+cos[m(θ−Δθ)+(ωt−Δφ)].

Or, factoring the anticlockwise H₊ and clockwise H⁻ components:

$\begin{matrix}{\mspace{79mu} {{H_{tot} = {H_{+} + {H\_}}}\mspace{79mu} {where}}} & (13) \\{H_{+} = {{{\cos \left( {{m\; \theta} - {\omega \; t}} \right)} + {\cos \left\lbrack {{m\left( {\theta - {\Delta \; \theta}} \right)} - \left( {{\omega \; t} - {\Delta \; \varphi}} \right)} \right\rbrack}} = {2\; {\cos \left\lbrack \frac{{2\; m\; \theta} - {2\; \omega \; t} - {m\; \Delta \; \theta} + {\Delta\varphi}}{2} \right\rbrack}{\cos \left( \frac{{m\; \Delta \; \theta} - {\Delta \; \varphi}}{2} \right)}}}} & (14) \\{\mspace{79mu} {{H\_} = {2\; {\cos \left( {{m\; \theta} + {\omega \; t} + \frac{{{- m}\; \Delta \; \theta} + {\Delta \; \varphi}}{2}} \right)}{{\cos \left( \frac{{m\; \Delta \; \theta} + {\Delta \; \varphi}}{2} \right)}.}}}} & (15)\end{matrix}$

Condition for the Clockwise Rotation for TE_(mnl) and TM_(mnl):

The anticlockwise term will vanish, if the last term of Eqn. (14) isnull, explicitly:

$\begin{matrix}{\frac{{m\; \Delta \; \theta} - {\Delta \; \varphi}}{2} = {{\frac{\pi}{2} + {k\; \pi \mspace{14mu} {for}\mspace{14mu} k}} = {{integer}.}}} & (16)\end{matrix}$

If the following condition as well as that of Eqn. (16) aresimultaneously satisfied,

$\begin{matrix}{\frac{{m\; \Delta \; \theta} + {\Delta \; \varphi}}{2} = {{\frac{\pi}{2} + {p\; \pi \mspace{14mu} {for}\mspace{14mu} p}} = {integer}}} & (17)\end{matrix}$

then, neither the anticlockwise nor clockwise waves are excited. Thissimultaneous condition can be provided by:

$\left\{ {\begin{matrix}{\frac{{m\; \Delta \; \theta} - {\Delta \; \varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k = {integer}} \\{{\Delta \; \varphi} = {q\; \pi}} & {q = {integer}}\end{matrix}.} \right.$

Conversely, the necessary and sufficient condition to excite only theclockwise rotation for TE_(nml) or TM_(nml) can be summarized as:

$\begin{matrix}\left\{ {\begin{matrix}{\frac{{m\; \Delta \; \theta} - {\Delta \; \varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k\text{:}\mspace{14mu} {integer}} \\{{\Delta \; \varphi} \neq {q\; \pi}} & {q\text{:}\mspace{14mu} {integer}}\end{matrix}.} \right. & (18)\end{matrix}$

To maximize the energy transfer efficiency of the clockwise rotation,the last term of Eqn. (15) must be ±1, simultaneously with Eqn. (16),namely

$\left\{ {\begin{matrix}{\frac{{m\; \Delta \; \theta} - {\Delta \; \varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k\text{:}\mspace{14mu} {integer}} \\{\frac{{m\; \Delta \; \theta} + {\Delta \; \varphi}}{2} = {p\; \pi}} & {p\text{:}\mspace{14mu} {integer}}\end{matrix}\quad} \right.$

which can be reduced to

$\begin{matrix}{{{\Delta \; \varphi} = {{{- \frac{\pi}{2}} + {\left( {p - k} \right)\pi \mspace{31mu} \Delta \; \theta}} = {\frac{1 + {2\left( {k + p} \right)}}{2\; m}\pi \mspace{31mu} k}}},{p\text{:}\mspace{14mu} {{integers}.}}} & (19)\end{matrix}$

Eqn. (19) is included as a special case of Eqn. (18). However, Eqn. (19)is preferable because of its maximum efficiency. A furthersimplification is given by setting k=p

$\begin{matrix}{{\Delta \; \varphi} = {{{- \frac{\pi}{2}}\mspace{31mu} \Delta \; \theta} = {\frac{1 + {4\; k}}{2\; m}\pi \mspace{31mu} k\text{:}\mspace{14mu} {{integer}.}}}} & (20)\end{matrix}$

Microwave dual injections to excite a purely clockwise rotation with themaximum efficiency are summarized as follows:

Case of TE₁₁₁:

$\begin{matrix}{{{{Port}\mspace{14mu} Q\mspace{14mu} {separated}\mspace{14mu} {from}\mspace{14mu} {Port}\mspace{14mu} P\mspace{14mu} {by}}\mspace{14mu} \pm \frac{\pi}{2}}{{{{Temporal}\mspace{14mu} {phase}\mspace{14mu} {delay}\text{:}}\mspace{14mu} - {\frac{\pi}{2}\mspace{14mu} \left( {{i.e.\mspace{14mu} {phase}}\mspace{14mu} {advance}} \right)}};}} & (21)\end{matrix}$

Case of TE₁₇₁:

$\begin{matrix}{{{Port}\mspace{14mu} Q\mspace{14mu} {separated}\mspace{14mu} {from}\mspace{14mu} {Port}\mspace{14mu} P\mspace{14mu} {by}\mspace{14mu} \frac{\pi}{4}\mspace{14mu} {or}\mspace{14mu} \frac{5\; \pi}{4}}{{{Temporal}\mspace{14mu} {phase}\mspace{14mu} {delay}\text{:}}\mspace{14mu} - {\frac{\pi}{2}\mspace{14mu} {\left( {{i.e.\mspace{14mu} {phase}}\mspace{14mu} {advance}} \right).}}}} & (22)\end{matrix}$

Condition for Anticlockwise Rotation for TE_(mnl) and TM_(mnl):

In the same manner, the necessary and sufficient condition to exciteonly the anticlockwise rotation for TE_(mnl) or TM_(mnl) can besummarized as:

$\begin{matrix}\left\{ {\begin{matrix}{\frac{{m\; \Delta \; \theta} + {\Delta \; \varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k\text{:}\mspace{14mu} {integer}} \\{{\Delta \; \varphi} \neq {q\; \pi}} & {q\text{:}\mspace{14mu} {integer}}\end{matrix}.} \right. & (23)\end{matrix}$

Eqn. (23) defines Δθ and Δθ as a function of the user-selected indicesm, n and 1 of the modes TE_(mnl) or TM_(mnl). To maximize the energyefficiency of the anticlockwise rotation, the last term of Eqn. (16)should be ±1, simultaneously with Eqn. (15), namely:

$\begin{matrix}\left\{ \begin{matrix}{\frac{{m\; \Delta \; \theta} + {\Delta \; \varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k\text{:}\mspace{14mu} {integer}} \\{\frac{{m\; \Delta \; \theta} - {\Delta \; \varphi}}{2} = {p\; \pi}} & {p\text{:}\mspace{14mu} {integer}}\end{matrix} \right. & (24)\end{matrix}$

which can be reduced to

$\begin{matrix}{{{\Delta \; \varphi} = {{\frac{\pi}{2} + {\left( {k - p} \right)\pi \mspace{31mu} \Delta \; \theta}} = {\frac{1 + {2\left( {k + p} \right)}}{2m}\pi \mspace{31mu} k}}},{p\text{:}\mspace{14mu} {integers}}} & (25)\end{matrix}$

Or, a simplification by setting k=p gives

$\begin{matrix}{{\Delta \; \varphi} = {{\frac{\pi}{2}\mspace{31mu} \Delta \; \theta} = {\frac{1 + {4k}}{2m}\pi \mspace{31mu} k\text{:}\mspace{14mu} {{integer}.}}}} & (26)\end{matrix}$

Microwave dual injections to excite a purely clockwise rotation with themaximum efficiency are summarized as follows.

Case of TE₁₁₁:

$\begin{matrix}{{Port}\mspace{14mu} Q\mspace{14mu} {separated}\mspace{14mu} {from}\mspace{14mu} {Port}\mspace{14mu} P\mspace{14mu} {by}\mspace{14mu} \text{±}\frac{\pi}{2}} & (27) \\{{{Temporal}\mspace{14mu} {phase}\mspace{14mu} {delay}} = \frac{\pi}{2}} & \;\end{matrix}$

Case of TE₁₂₁:

$\begin{matrix}{{Port}\mspace{14mu} Q\mspace{14mu} {separated}\mspace{14mu} {from}\mspace{14mu} {Port}\mspace{14mu} P\mspace{14mu} {by}\mspace{14mu} \frac{\pi}{4}\mspace{14mu} {or}\mspace{14mu} \frac{5\pi}{4}} & (28) \\{{{Temporal}\mspace{14mu} {phase}\mspace{14mu} {delay}} = {\frac{\pi}{2}.}} & \;\end{matrix}$

Each one of Eqns. 18-20 and 23-26 defines Δθ and ΔØ as a function of theuser-selected indices m, n and 1 of the modes TE_(mnl) or TM_(mnl).

In summary, a rotating microwave is established in the cavity 120 forany resonant mode TE_(mnl) or TM_(mnl) of the cavity, where the user isfree to choose the values of the mode indices n, m and 1. This isaccomplished by setting the temporal phase difference ΔØ and theazimuthal angle Δθ between the ports P and Q as functions of m, n and 1,defined in an applicable one of the Eqns. 18-20 and 23-26. The foregoingis illustrated as a method in a block diagram of FIG. 3A. In FIG. 3A, aplasma reactor including a cylindrical microwave cavity overlying aworkpiece processing chamber is provided as in FIG. 3. First and secondinput ports P and Q are provided in a sidewall of the cylindricalmicrowave cavity spaced apart by an offset angle no, (block 600 of FIG.3A). A next step is to generate rotating microwaves of mode TE_(mnl) orTM_(mnl) in the cylindrical microwave cavity, wherein at least two of m,n and 1 are user-selected values of a TE or TM mode (block 602). This isdone by introducing into the cylindrical microwave cavity, throughrespective ones of the first and second coupling apertures, respectivemicrowave signals separated by a temporal phase difference ΔØ (block 604of FIG. 3A). The method includes adjusting values of the offset angle Δθand the temporal phase difference ΔØ to values which are a function ofat least two of the user-selected TE or TM mode indices m, n and 1, soas to produce rotating microwaves of mode TE_(mnl) or TM_(mnl) in thecylindrical microwave cavity (block 606 of FIG. 3A).

Generalized Amplitude Modulation for a Slow Rotation of TE_(mnl) andTM_(mnl) Mode in a Cylindrical Cavity:

FIG. 3 depicts a modification of the embodiment of FIG. 2 for amplitudemodulation for a slow rotation of TE_(mnl) and TM_(mnl) mode in acylindrical cavity. It is the same as that of FIG. 2 except for theabsence of monitoring antennas and absence of a signal feedbackcontroller.

Amplitude Modulations Radiated from Ports P and Q:

Microwave fields radiated from Ports P and Q, where P and Q arespatially separated by 90 degrees, should have the following forms ofamplitude modulation to make a slow rotation of frequency Ω_(a) on theorder of 1-1000 Hz:

ζ_(Px) =a cos(Ω_(a) t)cos(ωt+φ_(h))   (29)

ζ_(Qy) =±a sin(Ω_(a) t)cos(ωt+φ_(h))   (30)

where α is an arbitrary constant, Ω_(a) is an angular frequency ofrotation, t is a time, and φ_(h) is an arbitrary initial phase, and theplus and minus signs of Eqn. (30) correspond to anticlockwise andclockwise rotations, respectively. Then, an excited wave in acylindrical cavity can be represented by using an azimuthal angle θ:

η=2c cos(θ∓Ω_(a) t)cos(ωt+φ_(h))=[2c cos(Ω_(a) t)cos θ+{±2c sin(Ω_(a)t)}sin θ]cos(ωt+φ_(h))   (31)

When rewriting Eqns. (29)-(30) in x-y coordinate system, it can bestated: a vector input

{right arrow over (ζ)}={a cos(Ω_(a) t){circumflex over (x)}±a sin(Ω_(a)t)ŷ} cos(ωt+φ_(h))   (32)

excites a vector wave of

{right arrow over (η)}={2c cos(Ω_(a) t){circumflex over (x)}±2csin(Ω_(a) t)ŷ} cos(ωt+φ_(h))   (33)

where {circumflex over (x)} and ŷ are unit base vectors in x and ydirections, respectively.

In FIG. 3, the Ports P and Q are not necessarily located at a 90 degreeinterval. However, it is required that an excited wave should have theform of Eqn. (33). This problem can be converted to a coordinatetransformation from an orthogonal x-y system to an oblique a-b system asshown in FIG. 4.

In FIG. 4, a general vector P is defined as

{right arrow over (P)}=p{circumflex over (x)}+qŷ=râ+s{circumflex over(b)}  (34)

where the base vectors in the a-b system are defined as

â=a _(x) {circumflex over (x)}+a _(y) ŷ  (35)

{circumflex over (b)}=b _(x) {circumflex over (x)}+b _(y) ŷ  (36)

Hence, when the ports P and Q are separated by 90 degrees, Eqn. (33) canbe represented by

{right arrow over (P)}=a cos Ω_(a) t{circumflex over (x)}±a sin Ω_(a)tŷ  (36-2)

where the common temporal term cos(ωt+φ_(h)) has been skipped.

Thus, p and q in Eqn. (34) are defined as:

p=a cos Ω_(a)t   (36-3)

q=±a sin Ω_(a)t   (36-4)

To obtain the expression in the oblique system, let the reciprocal bases{circumflex over (α)} and {circumflex over (β)} correspond to the basesâ and {circumflex over (b)}

$\begin{matrix}{\hat{\alpha} = \frac{\hat{b} \times \hat{z}}{\hat{z} \cdot \left( {\hat{a} \times \hat{b}} \right)}} & \left( {36\text{-}5} \right) \\{\hat{\beta} = \frac{\hat{z} \times \hat{a}}{\hat{z} \cdot \left( {\hat{a} \times \hat{b}} \right)}} & \left( {36\text{-}6} \right)\end{matrix}$

where â and {circumflex over (b)} are defined as

â=a _(x) {circumflex over (x)}+a _(y) ŷ  (36-7)

{circumflex over (b)}=b _(x) {circumflex over (x)}+b _(y) ŷ  (36-8)

Multiplying (36-5) and (36-6) on the second and third terms of Eqn.(34), the coordinate transformation is obtained

$\begin{matrix}{r = \frac{{pb}_{y} - {qb}_{x}}{V}} & (37) \\{s = \frac{{- {pa}_{y}} - {qa}_{x}}{V}} & (38) \\{{{where}\mspace{14mu} V} = {{a_{x}b_{y}} - {a_{y}{b_{x}.}}}} & (39)\end{matrix}$

The coordinates of x-y system in Eqns. (32) and (33) are now transformedinto those of an a-b system, as follows: Inserting Eqns. (36-3) and(36-4) into (37), (38), an explicit form is obtained:

$\begin{matrix}{r = \frac{{\alpha \mspace{14mu} \cos \; \Omega_{a}{tb}_{y}} \mp {\alpha \mspace{14mu} \sin \; \Omega_{a}{tb}_{x}}}{V}} & (40) \\{s = \frac{{{- \alpha}\mspace{14mu} \cos \; \Omega_{a}{ta}_{y}} \pm {\alpha \mspace{14mu} \sin \; \Omega_{a}{ta}_{x}}}{V}} & (41)\end{matrix}$

In summary, when the Ports P and Q are spaced apart with a general angledefined by Eqns (36-7) and (36-8) as shown in FIG. 3 or 5, a slowlyrotating microwave field of rotation frequency of Ω_(a) can be excitedby microwave field inputs from Ports P and Q represented by:

ζ_(Pa) =r cos(ωt+φ_(h))   (42-1)

ζ_(Qb) =s cos(ωt+φ_(h))   (42-2)

where r and s are defined in Eqns. (40) and (41), and the plus and minussign of Eqn. (41) corresponds to anticlockwise and clockwise rotations,respectively. The forms of (42-1) and (42-2) are of the form ofamplitude modulation with time varying functions of r and s.

Relating to Eqns. (22) and (28), we shall illustrate the case that PortQ is separated from port P by

$\frac{5\pi}{4}$

to make a slow rotation of TE₁₂₁ as shown in FIG. 5:

$\begin{matrix}{\hat{a} = {{{a_{x}\hat{x}} + {a_{y}\hat{y}}} = {{1\hat{x}} + {0\hat{y}}}}} & (43) \\{\hat{b} = {{{b_{x}\hat{x}} + {b_{y}\hat{y}}} = {{\left( {- \frac{1}{\sqrt{2}}} \right)\hat{x}} + {\left( {- \frac{1}{\sqrt{2}}} \right){\hat{y}.}}}}} & (44)\end{matrix}$

Substitution of Eqns. (43) and (44) into Eqns. (39)-(41) yields:

r=a cos Ω_(a) t−a sin Ω_(a) t   (45)

s=−√{square root over (2)}a sin Ω_(a) t.   (46)

This shows that, for the geometrical configuration of FIG. 5, whensupplying microwave power in the form of Eqns. (45) and (46) from PortsP and Q respectively, the excited wave will be equal to that of Eqn.(33). This is verified by the fact that substitution of Eqns. (43)-(46)into Eqn. (34) yields Eqn. (36-2), which leads to Eqn. (32), eventuallyto Eqn. (33). For other configurations of Ports P and Q, the skilledworker can derive supplied powers of each port in the same manner as theforegoing.

Impedance Shifting by Irises in a Power-Supplying Waveguide:

Each of the two waveguides 360 of the embodiment of FIG. 2 or theembodiment of FIG. 3 has a radiation or coupling aperture 405 openthrough a respective one of the ports P and Q to the interior of thecylindrical cavity 120, as shown in FIG. 6. In the embodiment depictedin FIG. 6, the waveguide 360 is rectangular, having four conductivewalls forming a rectangular cross-section, including a pair of sidewalls 410, 411, a floor 412 and a ceiling 413. An input opening 415 ofthe waveguide 360 is open for receiving microwaves. An opposite end 416is covered by a wall 418. The coupling aperture 405 referred to above isformed in the wall 418 and is aligned with a corresponding one of theports P and Q. Each port P and Q is an opening in the side wall of thecavity 120 and may match the dimensions of the coupling aperture 405.

The waveguide 360 may include one or more irises such as an iris 420.The iris 420 is formed as a rectangular window in a rectangular wall422. Behavior of the waveguide 360 is determined by the dimensions ofthe rectangular input opening 415, a×b, the dimensions of therectangular iris 420, c×d, the dimensions of the rectangular couplingaperture 405, e×f, the distance g between the iris 420 and the input end415 and the distance h between the iris 420 and the coupling aperture405. Other suitable shapes and dimensions can be chosen. To tune achamber impedance, the coupling aperture size e×f is first adjusted. Inone example, the best spectrum of a resonance “1” was obtained fore×f=60 mm×2 mm. For brevity of explanation, only the resonance “1” willbe considered hereinafter.

Next, an arbitrary distance h of the iris 420 from the coupling aperture405 is chosen. In FIG. 7, a capacitive iris is chosen, and the value ofthe dimension d is adjusted. As the size of d is changed, the impedancesof three resonances move. Furthermore, its quality factor Q representedby the size of a resonant circle in a Smith chart also changes. FIG. 8depicts an inductive iris, with which different impedance shifts may beobtained. A resonant iris shown in FIG. 9 can make a critical couplingwhere the chamber impedance of a target frequency is located on thecenter of a Smith chart. In this configuration, the stub tuners of FIGS.2 and 3 can be removed, leading to a simplified and cost-effectivechamber design. However, since the critical coupling has a high Q value,this setting will generally deteriorate repeatability of tuning.

As indicated in dashed line, a second iris plate 500 can be placed inthe waveguide 360 to obtain a preferable chamber impedance. A third irisplate may be added as well.

Advantages:

A principal advantage of the embodiment of FIGS. 1-5 is that a rotatingmicrowave for plasma processing can be produced for any suitablecombination of user-selected mode indices m, n and 1 of modes TE_(mnl)and TM_(mnl) by suitable adjustment of spatial and temporal separationbetween microwave excitations at two different azimuthal locations. Aprincipal advantage of the embodiment of FIGS. 6-9 is that the microwavechamber impedance can be adjusted without changing the chamber byintroducing impedance shifting irises into the power-feeding waveguidescoupled to the cavity.

While the foregoing is directed to embodiments of the present invention,other and further embodiments of the invention may be devised withoutdeparting from the basic scope thereof, and the scope thereof isdetermined by the claims that follow.

1. In a plasma reactor comprising a cylindrical microwave cavityoverlying a workpiece processing chamber, and first and second inputports P and Q in a sidewall of said cylindrical microwave cavity spacedapart by an offset angle Δθ, a method of generating rotating microwavesof mode TE_(mnl) or TM_(mnl) in said cylindrical microwave cavity,wherein m, n and 1 are user-selected values of a TE or TM mode, saidmethod comprising: introducing into said cylindrical microwave cavity,through respective ones of said first and second coupling apertures,respective microwave signals separated by a temporal phase differenceΔØ; adjusting values of said offset angle Δθ and said temporal phasedifference ΔØ to values which are a function of at least two of saiduser-selected TE or TM mode indices m, n and 1 so as to produce rotatingmicrowaves of mode TE_(mnl) or TM_(mnl) in said cylindrical microwavecavity.
 2. The method of claim 1 wherein said function is defined as:$\quad\left\{ \begin{matrix}{\frac{{m\; {\Delta\theta}} - {\Delta\varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k\text{:}\mspace{14mu} {integer}} \\{{\Delta\varphi} \neq {n\; \pi}} & {n\text{:}\mspace{14mu} {integer}}\end{matrix} \right.$
 3. The method of claim 2 wherein said rotatingmicrowaves rotate clockwise with the rotation frequency equal to anoperational microwave frequency.
 4. The method of claim 1 wherein saidfunction is defined as: $\quad\left\{ \begin{matrix}{\frac{{m\; {\Delta\theta}} - {\Delta\varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k\text{:}\mspace{14mu} {integer}} \\{\frac{{m\; {\Delta\theta}} + {\Delta\varphi}}{2} = {l\; \pi}} & {l\text{:}\mspace{14mu} {integer}}\end{matrix} \right.$
 5. The method of claim 1 wherein said function isdefined as: $\quad\left\{ \begin{matrix}{\frac{{m\; {\Delta\theta}} + {\Delta\varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k\text{:}\mspace{14mu} {integer}} \\{{\Delta\varphi} \neq {n\; \pi}} & {n\text{:}\mspace{14mu} {integer}}\end{matrix} \right.$
 6. The method of claim 5 wherein said rotatingmicrowaves rotate anticlockwise with the rotation frequency equal to anoperational microwave frequency.
 7. The method of claim 1 wherein saidfunction is defined as: $\quad\left\{ \begin{matrix}{\frac{{m\; {\Delta\theta}} + {\Delta\varphi}}{2} = {\frac{\pi}{2} + {k\; \pi}}} & {k\text{:}\mspace{14mu} {integer}} \\{\frac{{m\; {\Delta\theta}} - {\Delta\varphi}}{2} = {p\; \pi}} & {p\text{:}\mspace{14mu} {integer}}\end{matrix} \right.$
 8. The method of claim 1 wherein a first one ofsaid respective microwave signals is of a form:H _(P) ∝ cos(η+mθ−ωt)+cos(η+mθ+ωt) where ω is an angular frequency ofthe respective microwave signals and t is time, and η=0 or$\frac{\pi}{2}.$
 9. The method of claim 8 wherein a second one of saidrespective microwave signals is of a form:H _(Q) ∝ cos[η+m(θ−Δθ)−(ωt−Δφ)]+cos[η+m(θ−Δθ)+(ωt−Δφ)] where ω is anangular frequency of the microwave signals and t is time, and η=0 or$\frac{\pi}{2}.$
 10. In a plasma reactor comprising a cylindricalmicrowave cavity overlying a workpiece processing chamber, and first andsecond input ports P and Q in a sidewall of said cylindrical microwavecavity spaced apart by a general angle, a method of generating rotatingmicrowaves in said cylindrical microwave cavity with rotation frequencyΩ_(a), said method comprising: setting said general angle to satisfy thefollowing Equations:â=a _(x) {circumflex over (x)}+a _(y) ŷ{circumflex over (b)}=b _(x) {circumflex over (x)}−b _(y) ŷ; inputtingto input ports P and Q microwave fields represented respectively by:ζ_(Pa) =r cos(ωt+φ_(h))ζ_(Qb) =s cos(ωt+φ_(h)) where r and s are defined in the followingequations: $\begin{matrix}{r = \frac{{\alpha \mspace{14mu} \cos \; \Omega_{a}{tb}_{y}} \mp {\alpha \mspace{14mu} \sin \; \Omega_{a}{tb}_{x}}}{V}} \\{s = \frac{{{- \alpha}\mspace{14mu} \cos \; \Omega_{a}{ta}_{y}} \pm {\alpha \mspace{14mu} \sin \; \Omega_{a}{ta}_{x}}}{V}}\end{matrix}$ and the sign ∓ in “r” determines whether the rotation isanticlockwise or clockwise.
 11. A plasma reactor comprising: acylindrical microwave cavity overlying a workpiece processing chamber,and first and second input ports, P and Q, in a sidewall of saidcylindrical microwave cavity spaced apart by an azimuthal angle; amicrowave source having a microwave frequency and having a pair ofmicrowave source outputs; a pair of respective waveguides, each of saidrespective waveguides having a microwave input end coupled to arespective one of said microwave source outputs and a microwave outputend coupled to a respective one of said first and second input ports; acoupling aperture plate at said output end, and a rectangular couplingaperture in said coupling aperture plate; an iris plate between saidcoupling aperture plate and said microwave input end, and a rectangulariris opening in said iris plate.
 12. The plasma reactor of claim 11wherein said rectangular coupling aperture and said rectangular irisopening have respective parallel axes along a long dimension of arespective one of said coupling aperture and said iris opening, saidrespective parallel axes being parallel to an axis of symmetry of saidcylindrical microwave cavity.
 13. The plasma reactor of claim 11 whereineach of said waveguides has a microwave propagation direction betweenthe microwave input end and the microwave output end, said microwavepropagation direction extending toward an axis of symmetry of saidcylindrical microwave cavity.
 14. The plasma reactor of claim 11wherein: said rectangular coupling aperture has long and shortdimensions e and f, respectively, corresponding to a user-selectedimpedance.
 15. The plasma reactor of claim 14 wherein: said rectangulariris opening has long and short dimensions c and d, respectively,corresponding to a user-selected resonance.
 16. The plasma reactor ofclaim 15 wherein said rectangular iris is a capacitive iris and has along dimension parallel to an axis of symmetry of said cylindricalmicrowave cavity.
 17. The plasma reactor of claim 15 wherein saidrectangular iris is an inductive iris and has a short dimension parallelto said axis of symmetry of said cylindrical microwave cavity.